Sub-Riemannian and sub-Lorentzian geometry on $\text{SU}(1,1)$ and on its universal cover

We study sub-Riemannian and sub-Lorentzian geometry on the Lie group $\text{SU}(1,1)$ and on its universal cover $\text{CSU}(1,1)$. In the sub-Riemannian case we find the distance function and completely describe sub-Riemannian geodesics on both $\text{SU}(1,1)$ and $\text{CSU}(1,1)$, connecting two fixed points. In particular, we prove that there is a strong connection between the conjugate loci and the number of geodesics. In the sub-Lorentzian case, we describe the geodesics connecting two points on $\text{CSU}(1,1)$, and compare them with Lorentzian ones. It turns out that the reachable sets for Lorentzian and sub-Lorentzian normal geodesics intersect but are not included one to the other. A description of the timelike future is obtained and compared in the Lorentzian and sub-Lorentzain cases.